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Integrability of the sub-Riemannian mean curvature of surfaces in the Heisenberg group


Authors: D. Danielli, N. Garofalo and D. M. Nhieu
Journal: Proc. Amer. Math. Soc. 140 (2012), 811-821
MSC (2010): Primary 49Q05; Secondary 53D10
Published electronically: November 2, 2011
MathSciNet review: 2869066
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Abstract: The problem of the local summability of the sub-Riemannian mean curvature $ \mathcal H$ of a hypersurface $ M$ in the Heisenberg group, or in more general Carnot groups, near the characteristic set of $ M$ arises naturally in several questions in geometric measure theory. We construct an example which shows that the sub-Riemannian mean curvature $ \mathcal H$ of a $ C^2$ surface $ M$ in the Heisenberg group $ \mathbb{H}^1$ in general fails to be integrable with respect to the Riemannian volume on $ M$.


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Additional Information

D. Danielli
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: danielli@math.purdue.edu

N. Garofalo
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: garofalo@math.purdue.edu

D. M. Nhieu
Affiliation: Department of Mathematics, National Central University, Jhongli City, Taoyuan County 32001, Taiwan, Republic of China
Email: dmnhieu@math.ncu.edu.tw

DOI: https://doi.org/10.1090/S0002-9939-2011-11058-X
Keywords: Minimal surfaces, $H$-mean curvature, integration by parts, first and second variation, monotonicity of the $H$-perimeter
Received by editor(s): August 25, 2010
Received by editor(s) in revised form: August 31, 2010
Published electronically: November 2, 2011
Additional Notes: The first author was supported in part by NSF grant CAREER DMS-0239771
The second author was supported in part by NSF Grant DMS-1001317
The third author was supported in part by NSC Grant 99-2115-M-008-013-MY3
Communicated by: Mario Bonk
Article copyright: © Copyright 2011 American Mathematical Society